Abstract
This paper proposes an approach to measure the extent of nonlinearity of the exposure of a financial asset to a given risk factor. The proposed measure exploits the decomposition of a conditional expectation into its linear and nonlinear components. We illustrate the method with the measurement of the degree of nonlinearity of a European style option with respect to the underlying asset. Next, we use the method to identify the empirical patterns of the return-risk trade-off on the SP500. The results are strongly supportive of a nonlinear relationship between expected return and expected volatility. The data seem to be driven by two regimes: one regime with a positive return-risk trade-off and one with a negative trade-off.
Highlights
Economic theories are often operationalized under linearity assumptions on the relationships between the underlying variables or joint normality assumptions on their distributions
Linearity and normality assumptions are needed in order to obtain analytical formulas and elegant characterizations of the phenomena of interest
Such assumptions may lead to wrong conclusions when they are not valid. Recognizing these limits and observing the relatively high frequency of extreme economic events, a body of the empirical literature in finance emphasizes the distributional characteristics of assets returns that do not reflect normality, in particular asymmetry and fat tails
Summary
Economic theories are often operationalized under linearity assumptions on the relationships between the underlying variables or joint normality assumptions on their distributions. I propose an approach to measure the degree of nonlinearity of the relationship between two variables. Unlike our proposed measure of nonlinearity, the Pearson linear correlation coefficient measures the propensity of a random variable Y of being replicated by a linear function of another variable X. A linear correlation coefficient lying strictly between −1 and 1 indicates that a fit of Y by a linear function of X will not be perfect This imperfection arises either from the dependence of Y on random factors other than X or from nonlinearity in the relationship between Y and X (or both).
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